Problem: Simplify the following expression: $y = \dfrac{-8x^2+25x- 18}{x - 2}$
Explanation: First use factoring by grouping to factor the expression in the numerator. This expression is in the form ${A}x^2 + {B}x + {C}$ First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-8)}{(-18)} &=& 144 \\ {a} + {b} &=& &=& {25} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $144$ and add them together. The factors that add up to ${25}$ will be your ${a}$ and ${b}$ When ${a}$ is ${9}$ and ${b}$ is ${16}$ $ \begin{eqnarray} {ab} &=& ({9})({16}) &=& 144 \\ {a} + {b} &=& {9} + {16} &=& 25 \end{eqnarray} $ Next, rewrite the expression as $({A}x^2 + {a}x) + ({b}x + {C})$ $ ({-8}x^2 +{9}x) + ({16}x {-18}) $ Factor out the common factors: $ x(-8x + 9) - 2(-8x + 9)$ Now factor out $(-8x + 9)$ $ (-8x + 9)(x - 2)$ The original expression can therefore be written: $ \dfrac{(-8x + 9)(x - 2)}{x - 2}$ We are dividing by $x - 2$ , so $x - 2 \neq 0$ Therefore, $x \neq 2$ This leaves us with $-8x + 9; x \neq 2$.